Glossary

2.2 Hausdorff dimension and its calculation

5 min read•august 14, 2024

is a powerful tool for measuring the "size" of sets, especially those with irregular shapes. It captures fractional dimensions and provides insights into scaling properties, making it crucial for analyzing complex structures like fractals.

Calculating Hausdorff dimension involves using and examining how sets scale. For self-similar sets, there's a simple formula based on the number of copies and scaling factor. This approach reveals fascinating dimensions for well-known fractals like the .

Hausdorff dimension and measure

Definition and properties

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  • Hausdorff dimension is a non-negative real number that provides a way to measure the "size" of a set, taking into account its scaling properties and irregularities
  • Defined using Hausdorff measure, which is a generalization of the concept of Lebesgue measure
    • For a set F⊂RnF \subset \mathbb{R}^n and a non-negative real number ss, the ss-dimensional Hausdorff measure Hs(F)H^s(F) is defined as the infimum of the sum of the ssth powers of the diameters of covers of FF by sets of diameter at most δ\delta, as δ→0\delta \to 0
  • Hausdorff dimension dim⁡H(F)\dim_H(F) of a set FF is the infimum of all s≥0s \geq 0 such that Hs(F)=0H^s(F) = 0, or equivalently, the supremum of all s≥0s \geq 0 such that Hs(F)=∞H^s(F) = \infty
  • Captures the notion of "fractional dimension" and can be non-integer for irregular sets (fractals)
    • Example: The Hausdorff dimension of the Cantor set is log⁡(2)/log⁡(3)≈0.631\log(2) / \log(3) \approx 0.631, which is non-integer

Relation to Hausdorff measure

  • Hausdorff dimension is closely related to Hausdorff measure
  • For a set FF, if s<dim⁡H(F)s < \dim_H(F), then Hs(F)=∞H^s(F) = \infty, and if s>dim⁡H(F)s > \dim_H(F), then Hs(F)=0H^s(F) = 0
  • At the critical value s=dim⁡H(F)s = \dim_H(F), the Hausdorff measure Hs(F)H^s(F) may be zero, infinite, or finite and positive
    • Example: For the Cantor set CC, Hs(C)=∞H^s(C) = \infty for s<log⁡(2)/log⁡(3)s < \log(2) / \log(3), Hs(C)=0H^s(C) = 0 for s>log⁡(2)/log⁡(3)s > \log(2) / \log(3), and Hs(C)=1H^s(C) = 1 for s=log⁡(2)/log⁡(3)s = \log(2) / \log(3)

Hausdorff dimension of self-similar sets

Calculating Hausdorff dimension

  • Self-similar sets are sets that exhibit similar patterns at different scales, and their Hausdorff dimension can often be calculated explicitly
  • For a self-similar set F⊂RnF \subset \mathbb{R}^n that is the union of NN copies of itself, each scaled by a factor of rr, the Hausdorff dimension dim⁡H(F)\dim_H(F) satisfies the equation N×rdim⁡H(F)=1N \times r^{\dim_H(F)} = 1
    • This equation can be solved for dim⁡H(F)\dim_H(F) using logarithms: dim⁡H(F)=log⁡(N)/log⁡(1/r)\dim_H(F) = \log(N) / \log(1/r)

Examples of self-similar sets

  • The Cantor set is the union of two copies of itself, each scaled by a factor of 1/31/3, so its Hausdorff dimension is log⁡(2)/log⁡(3)≈0.631\log(2) / \log(3) \approx 0.631
  • The Sierpiński triangle is the union of three copies of itself, each scaled by a factor of 1/21/2, so its Hausdorff dimension is log⁡(3)/log⁡(2)≈1.585\log(3) / \log(2) \approx 1.585
  • The Koch curve is the union of four copies of itself, each scaled by a factor of 1/31/3, so its Hausdorff dimension is log⁡(4)/log⁡(3)≈1.262\log(4) / \log(3) \approx 1.262
  • The Sierpiński carpet is the union of eight copies of itself, each scaled by a factor of 1/31/3, so its Hausdorff dimension is log⁡(8)/log⁡(3)≈1.893\log(8) / \log(3) \approx 1.893

Dimension spectrum and applications

Concept of dimension spectrum

  • The dimension spectrum is a function that assigns a dimension to each point of a set based on the local scaling properties around that point
  • Provides a more refined characterization of the set's irregularity and heterogeneity compared to the global Hausdorff dimension
  • Related to the concept of multifractal analysis, which studies the distribution of local scaling exponents in a set or measure

Applications

  • Analysis of turbulence: The dimension spectrum can help characterize the intermittency and multi-scale nature of turbulent flows
  • Financial time series: The dimension spectrum can reveal the presence of different scaling behaviors in financial data (stock prices, exchange rates) and aid in risk assessment
  • Medical images: The dimension spectrum can be used to quantify the complexity and heterogeneity of medical images (MRI, CT scans) and assist in diagnosis and treatment planning
  • Geophysical data: The dimension spectrum can help analyze the fractal properties of geophysical data (seismic data, satellite imagery) and improve the understanding of underlying geological processes

Hausdorff dimension vs other dimensions

Topological dimension

  • , such as the covering dimension or the inductive dimension, is based on the local properties of a set and takes integer values
  • Does not capture the set's scaling properties or irregularities
    • Example: The topological dimension of the Cantor set is 0, while its Hausdorff dimension is log⁡(2)/log⁡(3)≈0.631\log(2) / \log(3) \approx 0.631

Box-counting dimension

  • Box-counting dimension, also known as or , is based on the asymptotic behavior of the number of boxes needed to cover a set at different scales
  • Easier to estimate numerically than Hausdorff dimension but may not always coincide with it
    • Example: For the Cantor set, the box-counting dimension and Hausdorff dimension coincide, but for the Sierpiński carpet, the box-counting dimension is log⁡(8)/log⁡(3)≈1.893\log(8) / \log(3) \approx 1.893, while the Hausdorff dimension is log⁡(8)/log⁡(3)−log⁡(4)/log⁡(9)≈1.864\log(8) / \log(3) - \log(4) / \log(9) \approx 1.864

Packing dimension

  • Packing dimension is dual to Hausdorff dimension and is defined using the concept of packing measure
  • Provides an upper bound for the Hausdorff dimension and coincides with it for many sets
    • Example: For the Cantor set and the Sierpiński triangle, the packing dimension and Hausdorff dimension coincide

Comparison

  • Hausdorff dimension is the most refined among these notions, as it captures the set's scaling properties and irregularities, and takes into account the size of covering sets in its definition
  • Topological dimension ≤\leq Hausdorff dimension ≤\leq Packing dimension
  • Box-counting dimension and Hausdorff dimension may coincide for some sets but not always
    • Example: For the Cantor set, all four dimensions (topological, Hausdorff, box-counting, and packing) coincide, but for the Sierpiński carpet, they are different

Key Terms to Review (19)

Borel Sets: Borel sets are the smallest σ-algebra containing all open sets in a given topological space, allowing for the construction of measurable sets. They play a crucial role in measure theory and provide a foundation for defining Lebesgue measure, Hausdorff measure, and other concepts in geometric measure theory.
Borel-Cantelli Lemma: The Borel-Cantelli Lemma is a fundamental result in probability theory that deals with the convergence of events in terms of measure. It states that if a sequence of events has a finite measure and the sum of their probabilities converges, then the probability that infinitely many of these events occur is zero. This lemma plays a crucial role in understanding the relationships between events and their measures, particularly in the context of Hausdorff dimension where we analyze sets in terms of their size and structure.
Cantor Set: The Cantor set is a classic example of a fractal and a perfect set that is constructed by repeatedly removing the middle third of a line segment. This process creates a set that is uncountably infinite yet has a Lebesgue measure of zero, which illustrates the counterintuitive properties of size and dimensionality in mathematical analysis.
Countable Additivity: Countable additivity is a property of a measure that states if you have a countable collection of disjoint sets, the measure of their union is equal to the sum of their measures. This concept is fundamental in the study of measures, linking how we can quantify the size or volume of sets to the way those sets combine. Understanding countable additivity helps clarify essential properties of measures, especially in the context of more complex structures like Lebesgue measure and Hausdorff dimension.
Covering Method: The covering method is a technique used in geometric measure theory to estimate and calculate the Hausdorff dimension of a set. This approach involves covering a set with a collection of balls or other shapes of varying sizes, allowing for an analysis of how the number of coverings changes as the sizes of the covers shrink. This method provides insight into the 'size' and structure of sets that may not have a traditional geometric dimension, facilitating a deeper understanding of their properties.
David T. Cohen: David T. Cohen is a mathematician known for his contributions to geometric measure theory, particularly in relation to the Hausdorff dimension and its calculations. His work emphasizes the importance of understanding the intricate structure of sets and measures, linking geometric properties with measure-theoretic concepts. His research provides tools and insights that are crucial for calculating the Hausdorff dimension, which measures the 'size' of a fractal set in a rigorous way.
Dynamical Systems: Dynamical systems refer to mathematical models that describe how a point in a certain space evolves over time according to specific rules. These systems can be deterministic or stochastic and are essential in studying complex behaviors in various fields, such as physics, biology, and economics. In the context of Hausdorff dimension, dynamical systems often help analyze the properties of fractals and the intricacies of their geometric structures.
Embedding Dimension: The embedding dimension is the smallest integer 'n' such that a given topological space can be continuously mapped into a Euclidean space of dimension 'n'. This concept plays a crucial role in understanding how complex geometrical shapes can be represented within simpler structures, and it relates closely to the Hausdorff dimension, which measures the size of a fractal or irregular set. The embedding dimension helps reveal the inherent properties of spaces and provides insights into their geometric structure.
Felix Hausdorff: Felix Hausdorff was a prominent mathematician known for his contributions to topology and measure theory, particularly in defining the Hausdorff measure and dimension. His work laid the foundation for understanding geometric properties of sets in metric spaces, connecting measure theory to fractals and various dimensions of space, which is crucial for studying the nature of geometric objects and their sizes.
Fractal Dimension: Fractal dimension is a mathematical concept that extends the notion of dimensionality to non-integer values, capturing the complexity of fractal shapes and sets. It quantifies how a fractal pattern fills space as you zoom in, providing insights into the scaling behavior of irregular structures. This concept connects with various ideas, including measuring sets in a more nuanced way than traditional Euclidean dimensions and linking to other important measures like Hausdorff and Lebesgue measures.
Hausdorff Dimension: Hausdorff dimension is a concept that extends the notion of dimensionality beyond integers, providing a way to measure sets that are too 'irregular' to fit traditional dimensions. It captures the complexity of a set's structure and is particularly useful for analyzing fractals, which often exhibit non-integer dimensions. This measurement plays a crucial role in understanding the relationship between various types of measures and helps in the study of geometric properties in more abstract settings.
Hausdorff Measure: Hausdorff measure is a mathematical concept used to define and generalize the notion of size or measure in metric spaces, particularly for sets that may be irregular or fragmented, such as fractals. It extends the idea of Lebesgue measure by considering coverings of sets with arbitrary small scales, allowing for the measurement of more complex geometric structures.
Image Processing: Image processing refers to the manipulation and analysis of digital images through various algorithms to enhance, extract, or analyze information. It is a crucial aspect of computer vision and plays a significant role in measuring geometric properties and understanding structures within images, making it relevant for understanding dimensions and shapes in mathematical contexts.
Measure-theoretic: Measure-theoretic refers to the framework of measure theory, which provides a systematic way to assign a quantitative value to subsets of a given space, particularly in terms of size or volume. This concept is crucial for understanding the rigorous mathematical treatment of areas such as integration, probability, and, notably, Hausdorff dimension, which measures the size of fractals and irregular sets in a way that goes beyond traditional notions of length or area.
Minkowski Dimension: The Minkowski dimension, also known as the box-counting dimension, is a way to quantify the complexity or 'size' of a set in a space, particularly when it comes to fractals and irregular shapes. This dimension measures how the number of smaller boxes needed to cover a set scales with the size of those boxes. It's closely tied to the concept of Hausdorff dimension, as both dimensions assess how a set fills space, but Minkowski dimension is more computationally accessible.
N-dimensional: The term 'n-dimensional' refers to a mathematical space that has 'n' degrees of freedom, meaning it can be described using 'n' coordinates. This concept is crucial in understanding various mathematical theories, especially in fields like geometry and physics, where spaces can have more than three dimensions. In geometric measure theory, the n-dimensional framework allows for the analysis of complex sets and their properties in higher dimensions.
Self-similarity: Self-similarity is a property of an object or set that exhibits a repeating pattern at different scales. This characteristic is often observed in fractals, where small portions of the structure resemble the whole, enabling an intricate connection to both Hausdorff dimension calculations and the exploration of fractal sets. Understanding self-similarity allows for a deeper insight into how dimensions can be non-integer and how complex geometrical shapes can be mathematically defined and measured.
Sieve Method: The sieve method is a mathematical technique used to estimate the size and distribution of sets, particularly in number theory and measure theory. It often involves filtering out elements from a set through various criteria, allowing for a more precise analysis of its properties, such as the Hausdorff dimension, which provides a way to measure the 'size' of a set in a more nuanced manner than traditional methods.
Topological Dimension: Topological dimension is a concept used to describe the minimum number of coordinates needed to specify points in a given space. It provides a way to classify spaces based on their structural properties and helps in understanding the behavior of various geometric objects. The connection between topological dimension and other concepts, such as Hausdorff dimension, plays a critical role in analyzing more complex structures like sub-Riemannian spaces, where traditional notions of dimension may not apply straightforwardly.
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